The page doesn't say why matrices need to have whole-number (really just integer) entries, but I'd suspect it's because bad approximations to non-integer rationals accumulate sufficiently to make the recurrence unrecognizable.
It says that the underlying action is on the torus (R/Z)^2. If the entries of the matrix are not integers, do we have a well defined action on the torus? It seems to me that the answer is no because Z^2 would not be invariant by the action.
> It says that the underlying action is on the torus (R/Z)^2. If the entries of the matrix are not integers, do we have a well defined action on the torus? It seems to me that the answer is no because Z^2 would not be invariant by the action.
One can still define a map by taking the fractional part of the matrix-vector product, but the resulting map won't be continuous (with respect to the topology of the torus). In addition, if one wants the map be a homeomorphism (continuous with continuous inverse) then the determinant must have absolute value 1.
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[ 4.3 ms ] story [ 28.3 ms ] threadAh, good point.
https://www.publishersweekly.com/9780316638111